
It follows from the inequalities \(\bar{P}^\top Q_1 \bar{P} - 2\bar{P} \succeq -Q_1^{-1}\)  and \(\bar{\Lambda }^\top Q_2 \bar{\Lambda } - 2\bar{\Lambda } \succeq -Q_2^{-1}\)  for any \(\bar{P} \in \mathbb {R}^{n_\zeta \times n_\zeta }\)  and \(\bar{\Lambda } \in \mathbb {R}^{n_\phi \times n_\phi }\)  [1]}, [2]} that (REF ) implies
\(\begin{bmatrix}\mathcal {A} & \mathcal {B} \\ \mathcal {C} & \mathcal {D} \end{bmatrix}^\top \begin{bmatrix}Q_1^{-1} & 0 \\ 0 & Q_2^{-1} \end{bmatrix} \begin{bmatrix}\mathcal {A} & \mathcal {B} \\ \mathcal {C} & \mathcal {D} \end{bmatrix} - \begin{bmatrix}\rho ^2 Q_1^{-1} & 0 \\ 0 & Q_2^{-1} \end{bmatrix} \preceq 0.\) 
